Flood Routing by Gravitational Search Algorithm and Quantify Parameters Uncertainty of Nonlinear Muskingum Model

Document Type : Research Article


1 Shahid Bahonar University of Kerman



The occurrence of flood in the human’s history has always been one of mankind’s concerns. The methods of confronting this destructive phenomenon are of utter importance between researchers. One of the categories against this issue is flood routing. The financial losses of flood to human societies have made it very important to predict the occurrence of floods, so that it is necessary to accurately predict flood. In order to predict the outflows, in fact, the extraction of flood hydrographs is required in the downstream. The routing methods are divided into two hydraulic and hydrological groups. Hydraulic methods require the historical data and the solution of equations mainly through complex hydraulic methods is time consuming, however hydrological methods are preferred because of simplicity of their relative concepts. They are easy to implement and economize time. It is believed to be popular with researchers and has always tried to improve the accuracy of the results of the hydrological methods, which has become a good alternative to hydraulic methods. The Muskingum method is the most widely used hydrological routing technique which is divided into two groups of linear Muskingum and nonlinear Muskingum, depending on the relationship between the amount of storage and the inflows and outflows.
Various methods for estimating the hydrological parameters of Muskingum model have been presented. Techniques for estimating the parameters of the Muskingum model can be classified into three categories: mathematical techniques, phenomenon-mimicking techniques and hybrid algorithms. Among various methods of routing, three parameters nonlinear Muskingum method are hugely popular. Evolutionary algorithms are used to estimate the optimal parameters of the nonlinear Muskingum method because of their convergence rate, no need to make very accurate initial estimate of the hydrological parameters and their randomness nature. In this paper, a gravitational search algorithm which is based on the Kepler algorithm was first used to routing three different hydrographs. In fact, Kepler algorithm is inspired by the elliptical motion of planets around the sun. At different times, the planets are very close to the sun, which represent the stage of the exploration of the algorithm, and at other times the planets are far away from the sun and express the stage of exploitation of the algorithm.
Results and discussion
Using the combination of gravitational search algorithm and Kepler algorithm (GSA-Kepler), the parameters of the Muskingum model are calculated for routing three different hydrographs: Wilson (1974), Wye River and Veissman and Lewis (2003). The first example is a benchmark problem that was first considered by Wilson (1974) to estimate the parameters of the Muskingum model. This river has no branch to the Belmont and has very little flow. The results of the GSA-Kepler and the Segmented Least Squares Method, BFGS, HJ + DFP, HJ + CG, Genetic Algorithm, Immune Clonal Selection Algorithm, Harmony Search Algorithm and Free Parameter Setting Harmony Search Algorithm are compared with each other. The second example is the flood hydrograph in the Wye River. It has no tributaries from Erwood to Belmont and has very little lateral flow. The third model is a multi-peak flow hygrograph that was first studied by Veissman and Lewis (2003). For the second example, the results of the GSA-Kepler algorithm, COBSA, PSO, DE, GA, BFGS and WOA are showed and for the third example, the results of the GSA-Kepler are compared with the results of the WOA and MHBMO algorithms. After determining the optimal hydrologic parameters, their uncertainty is estimated using the possibility theory. Selecting an analysis of uncertainty depends on many factors, such as knowledge of uncertainty sources and model complexities. There is no definite guideline for choosing the specific uncertainty analysis method that works best. The principles of analyzing the possibility theory are based on fuzzy theory, which was first pronounced by Zadeh in 1965. To investigate the uncertainty of the nonlinear Muskingum model parameters based on the possibility theory, the aforementioned algorithm and other algorithms include Least Squares Method, Gravitational Search Algorithm, BFGS algorithm, HJ + DFP, HJ + CG, Genetic Algorithm, Immune Clonal Selection Algorithm, Harmony Search Algorithm and Free Parameter Setting Harmony Search Algorithm were used. Then three triangular membership functions were assigned to the hydrological variables and the uncertainty of these parameters was calculated using the fuzzy alpha cut method.
Comparing the results of the GSA-Kepler with the results of the previous studies shows that the combined algorithm used in this study has an acceptable accuracy and high convergence rate. Based on the fuzzy alpha cut method, it is determined that for Wilson (1974) the uncertainty of parameter k is greater than the uncertainty of parameters x and m.
Keywords: Membership function, GSA-Kepler, Possiblity theory.


Barati, R. (2011). Parameter estimation of nonlinear Muskingum models using Nelder-Mead simplex algorithm. Journal of Hydrologic Eng., 16(11), 946-954.
Barati, R. (2013). Application of excel solver for parameter estimation of the nonlinear Muskingum models. KSCE Journal of Civil Engineering, 17(5), 1139-1148.
Chu, H.J. and Chang, L.C. (2009). Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model. Journal of Hydrologic Engineering, 14(9), 1024-1027.
Das, A. (2004). Parameter estimation for Muskingum models. Journal of Irrigation and Drainage Engineering, 130(2), 140-147.
Geem, Z.W. (2006). Parameter estimation for the nonlinear Muskingum model using the BFGS technique. Journal of irrigation and drainage engineering, 132(5), 474-478.
Geem, Z.W. (2011). Parameter estimation of the nonlinear Muskingum model using parameter-setting-free harmony search. Journal of Hydrologic Engineering, 16(8), 684-688.
Gill, M.A. (1978). Flood routing by the Muskingum method. Journal of hydrology, 36(3-4), 353-363.
Hamedi, F., Bozorg-Haddad, O., Pazoki, M., Asgari, H.R., Parsa, M. and Loaiciga, H.A. (2016). Parameter estimation of extended nonlinear Muskingum models with the weed optimization algorithm. Journal of Irrigation and Drainage Engineering, 142(12), 04016059.
Kang, L. and Zhang, S. (2016). Application of the elitist-mutated PSO and an improved GSA to estimate parameters of linear and nonlinear Muskingum flood routing models. PLOS one, 11(1), e0147338.
Karahan, H., Gurarslan, G. and Geem, Z.W. (2012). Parameter estimation of the nonlinear Muskingum flood-routing model using a hybrid harmony search algorithm. Journal of Hydrologic Engineering, 18(3), 352-360.
Kim, J.H., Geem, Z.W. and Kim, E.S. (2001. Parameter estimation of the nonlinear Muskingum model using harmony search. JAWRA Journal of the American Water Resources Association, 37(5), 1131-1138.
Luo, J. and Xie, J. (2010). Parameter estimation for nonlinear Muskingum model based on immune clonal selection algorithm. Journal of Hydrologic Engineering, 15(10), 844-851.
McCarthy, G. T. (1938). The unit hydrograph and flood routing. Conf. of North Atlantic Division, U.S. Army Corps of Engineers, Rhode Island.
Moghaddam, A., Behmanesh, J. and Farsijani, A. (2016). Parameters estimation for the new four-parameter nonlinear Muskingum model using the particle swarm optimization. Water resources management, 30(7), 2143-2160.
Mohan, S. (1997). Parameter estimation of nonlinear Muskingum models using genetic algorithm. Journal of hydraulic engineering, 123(2), 137-142.
Najafi, R. and Hessami, M.R. (2017). Uncertainty modeling of statistical downscaling to assess climate change impacts on temperature and precipitation. Water resources management, 31(6), 1843-1858.
Niazkar, M. and Afzali, S.H. (2014). Assessment of modified honey bee mating optimization for parameter estimation of nonlinear Muskingum models. Journal of Hydrologic Engineering, 20(4), 04014055.
Niazkar, M. and Afzali, S.H. (2016). Application of new hybrid optimization technique for parameter estimation of new improved version of Muskingum model. Water resources management, 30(13), 4713-4730.
O'Donnell, T. (1985). A direct three-parameter Muskingum procedure incorporating lateral inflow. Hydrological Sciences Journal, 30(4), 479-496.
Raina, R. and Thomas, M. (2012). Fuzzy vs. probabilistic techniques to address uncertainty for radial distribution load flow simulation. Energy and Power Engineering, 4(02), 99.
Rashedi, E., Nezamabadi-Pour, H. and Saryazdi, S. (2009). GSA: a gravitational search algorithm. Information sciences, 179(13), 2232-2248.
Sarafrazi, S., Nezamabadi-pour, H. and Seydnejad, S. R. (2015). A novel hybrid algorithm of GSA with Kepler algorithm for numerical optimization. Journal of King Saud University-Computer and Information Sciences, 27(3), 288-296.
Tung, Y.K. (1985). River flood routing by nonlinear Muskingum method. Journal of hydraulic engineering, 111(12), 1447-1460.
Varón-Gaviria, C.A., Barbosa-Fontecha, J. L. and Figueroa-García, J. C. (2017). Fuzzy uncertainty in random variable generation: An α-cut approach. International Conference on Intelligent Computing, pp. 264-273. Springer, Cham.
Viessman, W. and Lewis, G.L. (2003). Introduction to hydrology, 5th Ed, New Delhi, India.
Wilson, E., M. (1974). Engineering hydrology, Hampshire, United Kingdom: Macmillan Education.
Xu, D.M., Qiu, L. and Chen, S.Y. (2012). Estimation of nonlinear Muskingum model parameter using differential evolution. Journal of Hydrologic Engineering, 17(2), 348-353.
Yoo, C., Lee, J. and Lee, M. (2017). Parameter Estimation of the Muskingum Channel Flood-Routing Model in Ungauged Channel Reaches. Journal of Hydrologic Engineering, 22(7), 05017005.
Yoon, J. and Padmanabhan, G. (1993). Parameter estimation of linear and nonlinear Muskingum models. Journal of Water Resources Planning and Management, 119(5), 600-610.
Yuan, X., Wu, X., Tian, H., Yuan, Y. and Adnan, R.M. (2016). Parameter identification of nonlinear Muskingum model with backtracking search algorithm. Water resources management, 30(8), 2767-2783.
Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338-353.
  • Receive Date: 07 January 2019
  • Revise Date: 25 August 2019
  • Accept Date: 13 September 2019
  • First Publish Date: 22 November 2019